Textbook Errors, 120
C. E. Nordman and S. M. Blinder
University of Michigan Ann Arbor, 48104
Collision Theory of Chemical Reactions
For almost every chemical reaction, the rate constant depends on temperature in accordance with Arrhenius’
empirical formula
k
Ae-EJRT
=
—
(2)
productjs
the Arrhenius formula can be understood on the basis of a simple collision model. Molecules A and B are idealized as spheres of diameter dA and dnj respectively, and a collision is said to occur whenever the centers of A and B approach within the collision diameter d One
can
also define
a
d.\ + da
=
2
-
wd1
(4)
By the simplest model, two molecules A and B will react if a collision between them occurs with a relative kinetic energy e equal to or exceeding the critical value to (= Ea/ No). Although the form of eqn. (1) is obtained on the basis of this theory, the magnitude of the experimental rate constant is overestimated by a factor typically of the order of 100. This is qualitatively rationalized as a steric effect, whereby only those collisions involving molecules in the proper relative orientation can result in reaction. A quantitatively correct collision-theory rate constant cannot therefore be calculated in any simple way. Perhaps as a consequence of this, most elementary physical chemistry texts present a spurious derivation of the Arrhenius formula. It is supposed that the reaction rate r—the number of molecules of A (or B) reacting per unit time per unit volume—is equal simply to the product of the collision frequency per unit volume, Zab, and the fraction of molecules with the requisite energy, f(to) r
=
ZAE/(t0)
-
ovJ&iAnB
(6)
Suggestions of material suitable for this column and guest colsuitable for publication directly should be sent with as many details as possible, and particularly with reference to modem textbooks, to W. H. Eberhardt, School of Chemistry, Georgia Institute of Technology, Atlanta, Georgia 30332. Since the purpose of this column is to prevent the spread and continuation of errors and not the evaluation of individual texts, the sources of errors discussed will not be cited. In order to be presented, an error must occur in at least two independent recent standard books. 1See, for example, Blinder, S. M., “Advanced Physical Chemistry,” The Macmillan Co., New York, 1969, p. 240-241.
/
Journal of Chemical Education
/ m V'2 4¶e~m'v'‘r2iT
=
p(vj)
i
=
A, B
(8)
It is possible to show that the associated distribution of relative speeds uAb is likewise Maxwellian1 P(cAB)
=
47rl,AB2(2^T’)
e
